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2019 Global fluctuations for 1D log-gas dynamics. Covariance kernel and support
Jeremie Unterberger
Electron. J. Probab. 24: 1-28 (2019). DOI: 10.1214/19-EJP288


We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,

\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]

with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta $-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.

We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.


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Jeremie Unterberger. "Global fluctuations for 1D log-gas dynamics. Covariance kernel and support." Electron. J. Probab. 24 1 - 28, 2019.


Received: 6 July 2018; Accepted: 2 March 2019; Published: 2019
First available in Project Euclid: 21 March 2019

zbMATH: 1412.60017
MathSciNet: MR3933200
Digital Object Identifier: 10.1214/19-EJP288

Primary: 60B20 , 60F05 , 60G20 , 60J60 , 60J75 , 60K35

Keywords: beta-ensembles , Dyson’s Brownian motion , Fluctuations , Hydrodynamic limit , log-gas , random matrices , Stieltjes transform , Support


Vol.24 • 2019
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